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Square Functions and Maximal Operators Associated with Radial Fourier Multipliers

Square Functions and Maximal Operators Associated with Radial Fourier Multipliers

Chapter:
(p.273) Chapter Twelve Square Functions and Maximal Operators Associated with Radial Fourier Multipliers
Source:
Advances in Analysis
Author(s):
Sanghyuk LeeKeith M. RogersAndreas Seeger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159416.003.0012

This chapter begins with an overview on square functions for spherical and Bochner–Riesz means which were introduced by Eli Stein, and discusses their implications for radial multipliers and associated maximal functions. It focuses on the Littlewood–Paley bounds for two square functions introduced by Stein, who had stressed their importance in harmonic analysis and many important variants and generalizations in various monographs. The chapter proves new endpoint estimates for these square functions, for the maximal Bochner–Riesz operator, and for more general classes of radial Fourier multipliers. The majority of the chapter is devoted to these proofs, such as for convolutions with spherical measures.

Keywords:   square functions, maximal operators, radial Fourier multipliers, Littlewood–Paley bounds, Bochner–Riesz operator, harmonic analysis, endpoint estimates, spherical measures

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